The one-loop contributions to the trilinear neutral gauge boson couplings $ZZV^\ast$ ($V=\gamma,Z,Z'$), parametrized in terms of one $CP$-conserving $f_5^{V}$ and one $CP$-violating $f_4^{V}$ form factors, are calculated in models with $CP$-violating flavor changing neutral current couplings mediated by the $Z$ gauge boson and an extra neutral gauge boson $Z'$. Analytical results are presented in terms of Passarino-Veltman scalar functions. Constraints on the vector and axial couplings of the $Z$ gauge boson $\left|g_{{VZ}}^{tu}\right|< 0.0096$ and $\left|g_{{VZ}}^{tc}\right|<0.011$ are obtained from the current experimental data on the $t\rightarrow Z q$ decays. It is found that in the case of the $ZZ\gamma^\ast$ vertex the only non-vanishing form factor is $f_5^{\gamma}$, which can be of the order of $10^{-3}$, whereas for the $ZZZ^\ast$ vertex both form factors $f_5^{Z}$ and $f_4^{Z}$ are non-vanishing and can be of the order of $10^{-6}$ and $10^{-5}$, respectively. Our estimates for $f_5^{\gamma}$ and $f_5^{Z}$ are smaller than those predicted by the standard model, where $f_4^{Z}$ is absent up to the one loop level. We also estimate the $ZZ{Z'}^{*}$ form factors arising from both diagonal and non-diagonal $Z'$ couplings within a few extension models. It is found that in the diagonal case $f_{5}^{Z'}$ is the only non-vanishing form factor and its real and imaginary parts can be of the order of $10^{-1}-10^{-2}$ and $ 10^{-2}-10^{-3}$, respectively, with the dominant contributions arising from the light quarks and leptons. In the non-diagonal case $f_{5}^{Z^\prime}$ can be of the order of $10^{-4}$, whereas $f_4^{Z'}$ can reach values as large as $10^{-7}-10^{-8}$, with the largest contributions arising from the $Z'tq$ couplings.

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